Technical Summary: Multi-objective Bayesian Optimization for Likelihood-Free Inference in Sequential Sampling Models

1. Problem Statement

In many scientific domains, particularly psychology and consumer behavior modeling, statistical models are defined by generative processes (simulators) that produce synthetic data. While natural for incorporating domain knowledge, these models often possess intractable likelihood functions, rendering traditional Bayesian inference impossible. Likelihood-Free Inference (LFI), or Simulation-Based Inference (SBI), addresses this by relying solely on model simulations.

A specific challenge arises when dealing with multi-source data (e.g., choice outcomes and response times in Sequential Sampling Models). Existing efficient LFI methods, such as Bayesian Optimization for Likelihood-Free Inference (BOLFI), typically require combining discrepancies from multiple data sources into a single scalar metric. This combination is often arbitrary; if not handled with extreme care, it can lead to:

• Information loss: The relative importance of different data sources may be obscured.
• Conservative uncertainty quantification: The resulting posterior may be overly diffuse.
• Inability to detect conflict: If data sources provide conflicting information (e.g., due to model misspecification), a single combined discrepancy may fail to explore the high-likelihood regions of all sources simultaneously.

Current structured density estimation methods (e.g., synthetic likelihood) often require excessive simulations, making them unsuitable for computationally expensive simulators.

2. Methodology: MOBOLFI

The authors propose Multi-objective Bayesian Optimization for Likelihood-Free Inference (MOBOLFI). This method extends the single-source BOLFI approach to handle multi-source data by treating the discrepancies of each data source as separate objectives in a multi-objective optimization framework.

Core Mechanism

1. Multi-dimensional Discrepancy: Instead of a single discrepancy $\Delta$, MOBOLFI defines a vector of discrepancies $\mathbf{D}(\theta) = (D_1(\theta), \dots, D_K(\theta))^\top$, where each $D_k$ represents the expected discrepancy between simulated and observed data for the $k$-th data source.
2. Surrogate Modeling: A multivariate Gaussian Process (GP) is used as a surrogate model for the vector-valued discrepancy function $\mathbf{D}(\theta)$. This model captures not only the uncertainty in each discrepancy but also the dependence structure (covariance) between the discrepancies of different data sources.
3. Multi-objective Bayesian Optimization (MOBO): The algorithm employs MOBO to sequentially select simulation parameters $\theta$. Specifically, it uses the Noisy Expected Hypervolume Improvement (NEHVI) acquisition function. This guides the search toward the "Pareto frontier"—the set of parameter values where the discrepancies for all data sources are simultaneously minimized (or where no single source can be improved without worsening another).
4. Likelihood Approximation: Once the GP surrogate is trained, it approximates the distribution of discrepancies. The likelihood for the joint data and for individual data sources is approximated using the cumulative distribution function (CDF) of a multivariate normal distribution derived from the GP predictions.
   • The joint likelihood is approximated as $p(\mathbf{S}_{obs}|\theta) \propto \Phi(\mathbf{t}; \boldsymbol{\mu}_n(\theta), \boldsymbol{\Sigma}_n(\theta) + \boldsymbol{\Sigma})$.
   • Marginal and conditional likelihoods for individual sources (e.g., $p(T_{obs}|\theta)$) are derived by decomposing the multivariate probability, allowing for separate inference on subsets of the data.

Implementation Details

• Tolerance Selection: The tolerance vector $\mathbf{t}$ is set based on the $q$-quantile of the training discrepancies (e.g., $q=0.05$).
• Scaling: To ensure numerical stability and effective exploration, discrepancies are scaled (e.g., using Mean Absolute Deviation) so that they operate on similar magnitudes before being fed into the MOBO algorithm.
• Posterior Sampling: The approximate likelihoods are used within Markov Chain Monte Carlo (MCMC) algorithms (specifically Differential Evolution MCMC) to sample from the approximate posterior without further expensive simulations.

3. Key Contributions

The paper outlines three primary contributions:

1. Efficient Exploration of Disjoint High-Likelihood Regions: MOBOLFI can explore high-likelihood regions for multiple data sources in a single run, even when these regions are disjoint (a scenario common in model misspecification). This is difficult for single-objective approaches which may get trapped in local optima of a combined metric.
2. Avoidance of Information Loss: By avoiding the need to combine discrepancies into a single scalar, MOBOLFI preserves the distinct information content of each data source. This prevents the "highly conservative" uncertainty quantification often seen when arbitrary weights are used to combine discrepancies.
3. Diagnostic Capabilities: The method enables the approximation of likelihoods for individual data sources. This allows researchers to:
   • Check for conflicting information between data sources (model checking).
   • Determine the relative importance of specific data sources for estimating individual parameters.

4. Experimental Results

The authors validate MOBOLFI through three examples:

A. Toy Example (Brownian Motion)

• Setup: A synthetic model with two data sources (Normal observations and discretized Brownian motion) estimating a 10-dimensional parameter.
• Findings:
  • MOBOLFI produced posteriors with lower variance and closer alignment to the true posterior compared to BOLFI with a weighted sum of discrepancies.
  • MOBOLFI successfully recovered the true posterior for individual data sources (conditioning on only one source) without additional computational cost after the joint run.
  • In a misspecified scenario (where the two data sources were generated by different true parameters), MOBOLFI identified the conflicting information, whereas BOLFI produced a posterior mode lying between the conflicting truths, failing to capture the distinct high-likelihood regions.

B. Synthetic MLBA (Multi-attribute Linear Ballistic Accumulator)

• Setup: An SSM used for decision-making, fitted to synthetic choice and response time (RT) data. The closed-form likelihood of MLBA served as a "gold standard."
• Findings:
  • MOBOLFI posteriors were significantly closer to the closed-form gold standard than BOLFI posteriors.
  • MOBOLFI demonstrated that Response Time (RT) data was crucial for estimating back-end parameters (decision threshold/difficulty), while Choice (CH) data was more informative for front-end parameters (preferences).
  • In misspecified scenarios, MOBOLFI explored the parameter space effectively, identifying local optima near the conflicting true parameters, whereas BOLFI failed to explore the full region of high likelihood for both sources simultaneously.

C. Real-World Data (Ride-hailing Drivers)

• Setup: An empirical study on Singapore ride-hailing drivers' preferences for electric vehicles (EVs), using choice and RT data.
• Findings:
  • MOBOLFI approximations were much closer to the closed-form likelihood estimates than BOLFI.
  • The analysis revealed that while choice data drove the estimation of preference parameters, RT data was essential for estimating the decision threshold.
  • Joint data provided more efficient estimates than marginal data sources alone.

D. Day Care Center Example (Appendix)

• Setup: Inference on bacterial transmission rates using summary statistics from multiple centers.
• Findings: MOBOLFI performed competitively with BOLFI even in this single-source context (when data summaries were partitioned), demonstrating robustness.

5. Significance and Claims

The paper claims that MOBOLFI offers a robust framework for Bayesian inference in complex, computationally expensive models with multiple data sources. Its significance lies in:

• Handling Misspecification: It provides a mechanism to detect and analyze conflicts between data sources, which is critical for trustworthy analysis in complex models like SSMs.
• Simulation Efficiency: It achieves accurate approximations with fewer simulations than methods requiring large batches of data for density estimation, making it suitable for expensive simulators.
• Interpretability: By separating likelihood contributions, it offers deeper insights into which data sources drive the estimation of specific parameters, a feature lost in single-discrepancy approaches.

The authors note that while MOBOLFI is motivated by SSMs, it is applicable to any LFI problem with multi-source data. They acknowledge challenges remain, such as the sensitivity to the choice of summary statistics and the initialization of the BO algorithm, suggesting these as areas for future research.